Computer-implemented system and method for correlating satellite imagery for use in photovoltaic fleet output estimation

ABSTRACT

A computer-implemented system and method for correlating satellite imagery for use in photovoltaic fleet output estimation is provided. Pixels in satellite imagery data of overhead sky clearness is correlated to a bounded area within a geographic region. Each pixel represents collective irradiance that is converted into point clearness indexes for the points within the bounded area relative to clear sky irradiance. The point clearness indexes in the point clearness indexes are averaged for the points within the bounded area into an area clearness index. A variance of the area clearness index is determined in proportion to a physical area covered by each pixel. For each point, a variance of the point clearness index is determined as a ratio of the area clearness index variance and the physical area relative to the point clearness index, regional cloud speed, and a time interval relating to a time resolution of collective irradiance observation.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation of U.S. patent applicationSer. No. 13/190,449, filed Jul. 25, 2011, U.S. Pat. No. 8,165,813,issued Apr. 24, 2012, the priority date of which is claimed and thedisclosure of which is incorporated by reference.

This invention was made with State of California support under AgreementNumber 722. The California Public Utilities Commission of the State ofCalifornia has certain rights to this invention.

FIELD

This application relates in general to photovoltaic power generationfleet planning and operation and, in particular, to acomputer-implemented system and method for correlating satellite imageryfor use in photovoltaic fleet output estimation.

BACKGROUND

The manufacture and usage of photovoltaic systems has advancedsignificantly in recent years due to a continually growing demand forrenewable energy resources. The cost per watt of electricity generatedby photovoltaic systems has decreased dramatically, especially whencombined with government incentives offered to encourage photovoltaicpower generation. Photovoltaic systems are widely applicable asstandalone off-grid power systems, sources of supplemental electricity,such as for use in a building or house, and as power grid-connectedsystems. Typically, when integrated into a power grid, photovoltaicsystems are collectively operated as a fleet, although the individualsystems in the fleet may be deployed at different physical locationswithin a geographic region.

Grid connection of photovoltaic power generation fleets is a fairlyrecent development. In the United States, the Energy Policy Act of 1992deregulated power utilities and mandated the opening of access to powergrids to outsiders, including independent power providers, electricityretailers, integrated energy companies, and Independent System Operators(ISOs) and Regional Transmission Organizations (RTOs). A power grid isan electricity generation, transmission, and distribution infrastructurethat delivers electricity from supplies to consumers. As electricity isconsumed almost immediately upon production, power generation andconsumption must be balanced across the entire power grid. A large powerfailure in one part of the grid could cause electrical current toreroute from remaining power generators over transmission lines ofinsufficient capacity, which creates the possibility of cascadingfailures and widespread power outages.

As a result, both planners and operators of power grids need to be ableto accurately gauge on-going power generation and consumption, andphotovoltaic fleets participating as part of a power grid are expectedto exhibit predictable power generation behaviors. Power production datais needed at all levels of a power grid to which a photovoltaic fleet isconnected, especially in smart grid integration, as well as by operatorsof distribution channels, power utilities, ISOs, and RTOs. Photovoltaicfleet power production data is particularly crucial where a fleet makesa significant contribution to the grid's overall energy mix.

A grid-connected photovoltaic fleet could be dispersed over aneighborhood, utility region, or several states and its constituentphotovoltaic systems could be concentrated together or spread out.Regardless, the aggregate grid power contribution of a photovoltaicfleet is determined as a function of the individual power contributionsof its constituent photovoltaic systems, which in turn, may havedifferent system configurations and power capacities. The systemconfigurations may vary based on operational features, such as size andnumber of photovoltaic arrays, the use of fixed or tracking arrays,whether the arrays are tilted at different angles of elevation or areoriented along differing azimuthal angles, and the degree to which eachsystem is covered by shade due to clouds.

Photovoltaic system power output is particularly sensitive to shadingdue to cloud cover, and a photovoltaic array with only a small portioncovered in shade can suffer a dramatic decrease in power output. For asingle photovoltaic system, power capacity is measured by the maximumpower output determined under standard test conditions and is expressedin units of Watt peak (Wp). However, at any given time, the actual powercould vary from the rated system power capacity depending upongeographic location, time of day, weather conditions, and other factors.Moreover, photovoltaic fleets with individual systems scattered over alarge geographical area are subject to different location-specific cloudconditions with a consequential affect on aggregate power output.

Consequently, photovoltaic fleets operating under cloudy conditions canexhibit variable and unpredictable performance. Conventionally, fleetvariability is determined by collecting and feeding direct powermeasurements from individual photovoltaic systems or equivalentindirectly derived power measurements into a centralized controlcomputer or similar arrangement. To be of optimal usefulness, the directpower measurement data must be collected in near real time at finegrained time intervals to enable a high resolution time series of poweroutput to be created. However, the practicality of such an approachdiminishes as the number of systems, variations in systemconfigurations, and geographic dispersion of the photovoltaic fleetgrow. Moreover, the costs and feasibility of providing remote powermeasurement data can make high speed data collection and analysisinsurmountable due to the bandwidth needed to transmit and the storagespace needed to contain collected measurements, and the processingresources needed to scale quantitative power measurement analysisupwards as the fleet size grows.

In addition, correlating overhead sky clearness between two locationsspanned by a large photovoltaic fleet can help relax the need to obtainpower measurements for every system in a photovoltaic fleet. Theindividual systems may be geographically dispersed and, in large scalephotovoltaic fleets, the photovoltaic arrays within each system couldsimilarly be dispersed. Estimating sky clearness for each photovoltaicsystem, though, is crucial to accurate power measurement quantification.Determining overhead sky clearness separately for each photovoltaicsystem can be needlessly time-consuming and inefficient in terms ofduplicative computational effort when a fleet includes photovoltaicsystems operating under nearly identical sky conditions.

For instance, one direct approach to obtaining high speed time seriespower production data from a fleet of existing photovoltaic systems isto install physical meters on every photovoltaic system, record theelectrical power output at a desired time interval, such as every 10seconds, and sum the recorded output across all photovoltaic systems inthe fleet at each time interval. The totalized power data from thephotovoltaic fleet could then be used to calculate the time-averagedfleet power, variance of fleet power, and similar values for the rate ofchange of fleet power. An equivalent direct approach to obtaining highspeed time series power production data for a future photovoltaic fleetor an existing photovoltaic fleet with incomplete metering and telemetryis to collect solar irradiance data from a dense network of weathermonitoring stations covering all anticipated locations of interest atthe desired time interval, use a photovoltaic performance model tosimulate the high speed time series output data for each photovoltaicsystem individually, and then sum the results at each time interval.

With either direct approach, several difficulties arise. First, in termsof physical plant, calibrating, installing, operating, and maintainingmeters and weather stations is expensive and detracts from cost savingsotherwise afforded through a renewable energy source. Similarly,collecting, validating, transmitting, and storing high speed data forevery photovoltaic system or location requires collateral datacommunications and processing infrastructure, again at possiblysignificant expense. Moreover, data loss occurs whenever instrumentationor data communications do not operate reliably.

Second, in terms of inherent limitations, both direct approaches onlywork for times, locations, and photovoltaic system configurations whenand where meters are pre-installed; thus, high speed time series powerproduction data is unavailable for all other locations, time periods,and photovoltaic system configurations. Both direct approaches alsocannot be used to directly forecast future photovoltaic systemperformance since meters must be physically present at the time andlocation of interest. Fundamentally, data also must be recorded at thetime resolution that corresponds to the desired output time resolution.While low time-resolution results can be calculated from high resolutiondata, the opposite calculation is not possible. For example,photovoltaic fleet behavior with a 10-second resolution can not bedetermined from data collected by existing utility meters that collectthe data with a 15-minute resolution.

The few solar data networks that exist in the United States, such as theARM network, described in G. M. Stokes et al., “The atmosphericradiation measurement (ARM) program: programmatic background and designof the cloud and radiation test bed,” Bulletin of Am. MeteorologicalSociety 75, 1201-1221 (1994), the disclosure of which is incorporated byreference, and the SURFRAD network, do not have high density networks(the closest pair of stations in the ARM network is 50 km apart) norhave they been collecting data at a fast rate (the fastest rate is 20seconds at ARM network and one minute at SURFRAD network).

The limitations of the direct measurement approaches have promptedresearchers to evaluate other alternatives. Researchers have installeddense networks of solar monitoring devices in a few limited locations,such as described in S. Kuszamaul et al., “Lanai High-Density IrradianceSensor Network for Characterizing Solar Resource Variability of MW-ScalePV System.” 35^(th) Photovoltaic Specialists Conf., Honolulu, Hi. (Jun.20-25, 2010), and R. George, “Estimating Ramp Rates for Large PV SystemsUsing a Dense Array of Measured Solar Radiation Data,” Am. Solar EnergySociety Annual Conf. Procs., Raleigh, N.C. (May 18, 2011), thedisclosures of which are incorporated by reference. As data are beingcollected, the researchers examine the data to determine if there areunderlying models that can translate results from these devices tophotovoltaic fleet production at a much broader area, yet fail toprovide translation of the data. In addition, half-hour or hourlysatellite irradiance data for specific locations and time periods ofinterest have been combined with randomly selected high speed data froma limited number of ground-based weather stations, such as described inCAISO 2011. “Summary of Preliminary Results of 33% Renewable IntegrationStudy—2010,” Cal. Public Util. Comm. LTPP Docket No. R.10-05-006 (Apr.29, 2011) and J. Stein, “Simulation of 1-Minute Power Output fromUtility-Scale Photovoltaic Generation Systems,” Am. Solar Energy SocietyAnnual Conf. Procs., Raleigh, N.C. (May 18, 2011), the disclosures ofwhich are incorporated by reference. This approach, however, does notproduce time synchronized photovoltaic fleet variability for anyparticular time period because the locations of the ground-based weatherstations differ from the actual locations of the fleet. While suchresults may be useful as input data to photovoltaic simulation modelsfor purpose of performing high penetration photovoltaic studies, theyare not designed to produce data that could be used in grid operationaltools.

Therefore, a need remains for an approach to efficiently estimatingpower output of a photovoltaic fleet in the absence of high speed timeseries power production data. As well, a further need remains for anapproach to correlating weather conditions experienced by differentcomponents of a geographically dispersed photovoltaic fleet.

SUMMARY

Approaches to generating high-speed time series photovoltaic fleetperformance data and to correlating point-to-point cloudy conditions aredescribed.

One embodiment provides a computer-implemented system and method forcorrelating satellite imagery for use in photovoltaic fleet outputestimation. A set of pixels in satellite imagery data of overhead skyclearness is correlated to a bounded area within a geographic region.Each pixel represents collective irradiance over a plurality of pointswithin the bounded area. Each point is suitable for operation of aphotovoltaic system included in a photovoltaic fleet. The collectiveirradiance is converted into a set of point clearness indexes for eachof the points within the bounded area relative to clear sky irradiance.The point clearness indexes in the set of point clearness indexes areaveraged for each of the points within the bounded area into an areaclearness index. A variance of the area clearness index is determined inproportion to a physical area within the bounded area covered by eachpixel. For each of the points within the bounded area, a variance of thepoint clearness index for that point is determined as a ratio of thevariance of the area clearness index and the physical area relative tothe point clearness index for that point, cloud speed over thegeographic region, and a time interval relating to a time resolution ofobservation for the collective irradiance.

Some of the notable elements of this methodology non-exclusivelyinclude:

(1) Employing a fully derived statistical approach to generatinghigh-speed photovoltaic fleet production data;

(2) Using a small sample of input data sources as diverse asground-based weather stations, existing photovoltaic systems, or solardata calculated from satellite images;

(3) Producing results that are usable for any photovoltaic fleetconfiguration;

(4) Supporting any time resolution, even those time resolutions fasterthan the input data collection rate; and

(5) Providing results in a form that is useful and usable by electricpower grid planning and operation tools.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram showing a computer-implemented method forcorrelating satellite imagery for use in photovoltaic fleet outputestimation in accordance with one embodiment.

FIG. 2 is a block diagram showing a computer-implemented system forcorrelating satellite imagery for use in photovoltaic fleet outputestimation in accordance with one embodiment.

FIG. 3 is a graph depicting, by way of example, ten hours of time seriesirradiance data collected from a ground-based weather station with10-second resolution.

FIG. 4 is a graph depicting, by way of example, the clearness index thatcorresponds to the irradiance data presented in FIG. 3.

FIG. 5 is a graph depicting, by way of example, the change in clearnessindex that corresponds to the clearness index presented in FIG. 4.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5.

FIGS. 7A-7B are photographs showing, by way of example, the locations ofthe Cordelia Junction and Napa high density weather monitoring stations.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds.

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance.

FIGS. 10A-10F are graphs depicting, by way of example, the sameinformation as depicted in FIGS. 9A-9F versus temporal distance.

FIGS. 11A-11F are graphs depicting, by way of example, the predictedversus the measured variances of clearness indexes using differentreference time intervals.

FIGS. 12A-12F are graphs depicting, by way of example, the predictedversus the measured variances of change in clearness indexes usingdifferent reference time intervals.

FIGS. 13A-13F are graphs and a diagram depicting, by way of example,application of the methodology described herein to the Napa network.

FIG. 14 is a graph depicting, by way of example, an actual probabilitydistribution for a given distance between two pairs of locations, ascalculated for a 1,000 meter×1,000 meter grid in one square meterincrements.

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

FIG. 16 is a graph depicting, by way of example, results generated byapplication of Equation (18).

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions.

FIG. 18 is a graph depicting, by way of example, results by applicationof the model.

DETAILED DESCRIPTION

Photovoltaic cells employ semiconductors exhibiting a photovoltaiceffect to generate direct current electricity through conversion ofsolar irradiance. Within each photovoltaic cell, light photons exciteelectrons in the semiconductors to create a higher state of energy,which acts as a charge carrier for electric current. A photovoltaicsystem uses one or more photovoltaic panels that are linked into anarray to convert sunlight into electricity. In turn, a collection ofphotovoltaic systems can be collectively operated as a photovoltaicfleet when integrated into a power grid, although the constituentphotovoltaic systems may actually be deployed at different physicallocations within a geographic region.

To aid with the planning and operation of photovoltaic fleets, whetherat the power grid, supplemental, or standalone power generation levels,high resolution time series of power output data is needed toefficiently estimate photovoltaic fleet power production. Thevariability of photovoltaic fleet power generation under cloudyconditions can be efficiently estimated, even in the absence of highspeed time series power production data, by applying a fully derivedstatistical approach. FIG. 1 is a flow diagram showing acomputer-implemented method 10 for estimating power data for aphotovoltaic power generation fleet in accordance with one embodiment.The method 10 can be implemented in software and execution of thesoftware can be performed on a computer system, such as furtherdescribed infra, as a series of process or method modules or steps.

Preliminarily, a time series of solar irradiance data is obtained (step11) for a set of locations representative of the geographic regionwithin which the photovoltaic fleet is located or intended to operate,as further described infra with reference to FIG. 3. Each time seriescontains solar irradiance observations electronically recorded at knowninput time intervals over successive time periods. The solar irradianceobservations can include irradiance measured by a representative set ofground-based weather stations (step 12), existing photovoltaic systems(step 13), satellite observations (step 14), or some combinationthereof. Other sources of the solar irradiance data are possible.

Next, the solar irradiance data in the time series is converted overeach of the time periods, such as at half-hour intervals, into a set ofclearness indexes, which are calculated relative to clear sky globalhorizontal irradiance. The set of clearness indexes are interpreted intoas irradiance statistics (step 15), as further described infra withreference to FIGS. 4-6. The irradiance statistics for each of thelocations is combined into fleet irradiance statistics applicable overthe geographic region of the photovoltaic fleet. A time lag correlationcoefficient for an output time interval is also determined to enable thegeneration of an output time series at any time resolution, even fasterthan the input data collection rate.

Finally, power statistics, including a time series of the powerstatistics for the photovoltaic fleet, are generated (step 17) as afunction of the fleet irradiance statistics and system configuration,particularly the geographic distribution and power rating of thephotovoltaic systems in the fleet (step 16). The resultant high-speedtime series fleet performance data can be used to predictably estimatepower output and photovoltaic fleet variability by fleet planners andoperators, as well as other interested parties.

The calculated irradiance statistics are combined with the photovoltaicfleet configuration to generate the high-speed time series photovoltaicproduction data. In a further embodiment, the foregoing methodology canbe may also require conversion of weather data for a region, such asdata from satellite regions, to average point weather data. Anon-optimized approach would be to calculate a correlation coefficientmatrix on-the-fly for each satellite data point. Alternatively, aconversion factor for performing area-to-point conversion of satelliteimagery data is described in commonly-assigned U.S. patent application,entitled “Computer-Implemented System and Method for EfficientlyPerforming Area-To-Point Conversion of Satellite Imagery forPhotovoltaic Power Generation Fleet Output Estimation,” Ser. No.13/190,449, filed Jul. 25, 2011, pending, the disclosure of which isincorporated by reference.

The high resolution time series of power output data is determined inthe context of a photovoltaic fleet, whether for an operational fleetdeployed in the field, by planners considering fleet configuration andoperation, or by other individuals interested in photovoltaic fleetvariability and prediction, such as described in commonly-assigned U.S.patent application, entitled “Computer-Implemented System and Method forEstimating Power Data for a Photovoltaic Power Generation Fleet,” Ser.No. 13/190,442, filed Jul. 25, 2011, pending, and U.S. patentapplication, entitled “Computer-Implemented System and Method forEfficiently Performing Area-To-Point Conversion of Satellite Imagery forPhotovoltaic Power Generation Fleet Output Estimation,” Ser. No.13/190,449, filed Jul. 25, 2011, pending, the disclosure of which isincorporated by reference. FIG. 2 is a block diagram showing acomputer-implemented system 20 for correlating satellite imagery for usein photovoltaic fleet output estimation in accordance with oneembodiment. Time series power output data for a photovoltaic fleet isgenerated using observed field conditions relating to overhead skyclearness. Solar irradiance 23 relative to prevailing cloudy conditions22 in a geographic region of interest is measured. Direct solarirradiance measurements can be collected by ground-based weatherstations 24. Solar irradiance measurements can also be inferred by theactual power output of existing photovoltaic systems 25. Additionally,satellite observations 26 can be obtained for the geographic region.Both the direct and inferred solar irradiance measurements areconsidered to be sets of point values that relate to a specific physicallocation, whereas satellite imagery data is considered to be a set ofarea values that need to be converted into point values, as furtherdescribed infra. Still other sources of solar irradiance measurementsare possible.

The solar irradiance measurements are centrally collected by a computersystem 21 or equivalent computational device. The computer system 21executes the methodology described supra with reference to FIG. 1 and asfurther detailed herein to generate time series power data 26 and otheranalytics, which can be stored or provided 27 to planners, operators,and other parties for use in solar power generation 28 planning andoperations. The data feeds 29 a-c from the various sources of solarirradiance data need not be high speed connections; rather, the solarirradiance measurements can be obtained at an input data collection rateand application of the methodology described herein provides thegeneration of an output time series at any time resolution, even fasterthan the input time resolution. The computer system 21 includes hardwarecomponents conventionally found in a general purpose programmablecomputing device, such as a central processing unit, memory, userinterfacing means, such as a keyboard, mouse, and display, input/outputports, network interface, and non-volatile storage, and execute softwareprograms structured into routines, functions, and modules for executionon the various systems. In addition, other configurations ofcomputational resources, whether provided as a dedicated system orarranged in client-server or peer-to-peer topologies, and includingunitary or distributed processing, communications, storage, and userinterfacing, are possible.

The detailed steps performed as part of the methodology described suprawith reference to FIG. 1 will now be described.

Obtain Time Series Irradiance Data

The first step is to obtain time series irradiance data fromrepresentative locations. This data can be obtained from ground-basedweather stations, existing photovoltaic systems, a satellite network, orsome combination sources, as well as from other sources. The solarirradiance data is collected from several sample locations across thegeographic region that encompasses the photovoltaic fleet.

Direct irradiance data can be obtained by collecting weather data fromground-based monitoring systems. FIG. 3 is a graph depicting, by way ofexample, ten hours of time series irradiance data collected from aground-based weather station with 10-second resolution, that is, thetime interval equals ten seconds. In the graph, the blue line is themeasured horizontal irradiance and the black line is the calculatedclear sky horizontal irradiance for the location of the weather station.

Irradiance data can also be inferred from select photovoltaic systemsusing their electrical power output measurements. A performance modelfor each photovoltaic system is first identified, and the input solarirradiance corresponding to the power output is determined.

Finally, satellite-based irradiance data can also be used. As satelliteimagery data is pixel-based, the data for the geographic region isprovided as a set of pixels, which span across the region andencompassing the photovoltaic fleet.

Calculate Irradiance Statistics

The time series irradiance data for each location is then converted intotime series clearness index data, which is then used to calculateirradiance statistics, as described infra.

Clearness Index (Kt)

The clearness index (Kt) is calculated for each observation in the dataset. In the case of an irradiance data set, the clearness index isdetermined by dividing the measured global horizontal irradiance by theclear sky global horizontal irradiance, may be obtained from any of avariety of analytical methods. FIG. 4 is a graph depicting, by way ofexample, the clearness index that corresponds to the irradiance datapresented in FIG. 3. Calculation of the clearness index as describedherein is also generally applicable to other expressions of irradianceand cloudy conditions, including global horizontal and direct normalirradiance.

Change in Clearness Index (ΔKt)

The change in clearness index (ΔKt) over a time increment of Δt is thedifference between the clearness index starting at the beginning of atime increment t and the clearness index starting at the beginning of atime increment t, plus a time increment Δt. FIG. 5 is a graph depicting,by way of example, the change in clearness index that corresponds to theclearness index presented in FIG. 4.

Time Period

The time series data set is next divided into time periods, forinstance, from five to sixty minutes, over which statisticalcalculations are performed. The determination of time period is selecteddepending upon the end use of the power output data and the timeresolution of the input data. For example, if fleet variabilitystatistics are to be used to schedule regulation reserves on a 30-minutebasis, the time period could be selected as 30 minutes. The time periodmust be long enough to contain a sufficient number of sampleobservations, as defined by the data time interval, yet be short enoughto be usable in the application of interest. An empirical investigationmay be required to determine the optimal time period as appropriate.

Fundamental Statistics

Table 1 lists the irradiance statistics calculated from time series datafor each time period at each location in the geographic region. Notethat time period and location subscripts are not included for eachstatistic for purposes of notational simplicity.

TABLE 1 Statistic Variable Mean clearness index μ_(Kt) Varianceclearness index σ_(Kt) ² Mean clearness index change μ_(ΔKt) Varianceclearness index change μ_(ΔKt) ²

Table 2 lists sample clearness index time series data and associatedirradiance statistics over five-minute time periods. The data is basedon time series clearness index data that has a one-minute time interval.The analysis was performed over a five-minute time period. Note that theclearness index at 12:06 is only used to calculate the clearness indexchange and not to calculate the irradiance statistics.

TABLE 2 Clearness Index (Kt) Clearness Index Change (ΔKt) 12:00 50% 40%12:01 90%  0% 12:02 90% −80%  12:03 10%  0% 12:04 10% 80% 12:05 90%−40%  12:06 50% Mean (μ) 57%  0% Variance (σ²) 13% 27%

The mean clearness index change equals the first clearness index in thesucceeding time period, minus the first clearness index in the currenttime period divided by the number of time intervals in the time period.The mean clearness index change equals zero when these two values arethe same. The mean is small when there are a sufficient number of timeintervals. Furthermore, the mean is small relative to the clearnessindex change variance. To simplify the analysis, the mean clearnessindex change is assumed to equal zero for all time periods.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5 using a half-hour hour time period.Note that FIG. 6 presents the standard deviations, determined as thesquare root of the variance, rather than the variances, to present thestandard deviations in terms that are comparable to the mean.

Calculate Fleet Irradiance Statistics

Irradiance statistics were calculated in the previous section for thedata stream at each sample location in the geographic region. Themeaning of these statistics, however, depends upon the data source.Irradiance statistics calculated from a ground-based weather stationdata represent results for a specific geographical location as pointstatistics. Irradiance statistics calculated from satellite datarepresent results for a region as area statistics. For example, if asatellite pixel corresponds to a one square kilometer grid, then theresults represent the irradiance statistics across a physical area onekilometer square.

Average irradiance statistics across the photovoltaic fleet region are acritical part of the methodology described herein. This section presentsthe steps to combine the statistical results for individual locationsand calculate average irradiance statistics for the region as a whole.The steps differs depending upon whether point statistics or areastatistics are used.

Irradiance statistics derived from ground-based sources simply need tobe averaged to form the average irradiance statistics across thephotovoltaic fleet region. Irradiance statistics from satellite sourcesare first converted from irradiance statistics for an area intoirradiance statistics for an average point within the pixel. The averagepoint statistics are then averaged across all satellite pixels todetermine the average across the photovoltaic fleet region.

Mean Clearness Index (μ _(Kt) ) and Mean Change in Clearness Index(μΔKt)

The mean clearness index should be averaged no matter what input datasource is used, whether ground, satellite, or photovoltaic systemoriginated data. If there are N locations, then the average clearnessindex across the photovoltaic fleet region is calculated as follows.

$\begin{matrix}{\mu_{\overset{\_}{Kt}} = {\sum\limits_{i = 1}^{N}\frac{\mu_{{Kt}_{i}}}{N}}} & (1)\end{matrix}$

The mean change in clearness index for any period is assumed to be zero.As a result, the mean change in clearness index for the region is alsozero.μ _(ΔKt) =0  (2)

Convert Area Variance to Point Variance

The following calculations are required if satellite data is used as thesource of irradiance data. Satellite observations represent valuesaveraged across the area of the pixel, rather than single pointobservations. The clearness index derived from this data (Kt^(Area)) maytherefore be considered an average of many individual pointmeasurements.

$\begin{matrix}{{Kt}^{Area} = {\sum\limits_{i = 1}^{N}\frac{\;_{{Kt}^{i}}}{N}}} & (3)\end{matrix}$

As a result, the variance of the clearness index based on satellite datacan be expressed as the variance of the average clearness index acrossall locations within the satellite pixel.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{{VAR}\left\lbrack {Kt}^{Area} \right\rbrack} = {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}\frac{{Kt}^{i}}{N}} \right\rbrack}}} & (4)\end{matrix}$

The variance of a sum, however, equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {{Kt}^{i},{Kt}^{j}} \right\rbrack}}}}} & (5)\end{matrix}$

Let ρ^(Kt) ^(i) ^(,Kt) ^(j) represents the correlation coefficientbetween the clearness index at location i and location j within thesatellite pixel. By definition of correlation coefficient,COV[Kt^(i),Kt^(j)]=σ_(Kt) ^(i)σ_(Kt) ^(j)ρ^(Kt) ^(i) ^(,Kt) ^(j) .Furthermore, since the objective is to determine the average pointvariance across the satellite pixel, the standard deviation at any pointwithin the satellite pixel can be assumed to be the same and equalsσ_(Kt), which means that σ_(Kt) ^(i)σ_(Kt) ^(j)=σ_(Kt) ² for alllocation pairs. As a result, COV[Kt^(i), Kt^(j)]=σ_(Kt) ²ρ^(Kt) ^(i)^(,Kt) ^(j) . Substituting this result into Equation (5) and simplify.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{\sigma_{Kt}^{2}\left( \frac{1}{N^{2}} \right)}{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{{Kt}^{i},{Kt}^{j}}}}}} & (6)\end{matrix}$

Suppose that data was available to calculate the correlation coefficientin Equation (6). The computational effort required to perform a doublesummation for many points can be quite large and computationallyresource intensive. For example, a satellite pixel representing a onesquare kilometer area contains one million square meter increments. Withone million increments, Equation (6) would require one trillioncalculations to compute.

The calculation can be simplified by conversion into a continuousprobability density function of distances between location pairs acrossthe pixel and the correlation coefficient for that given distance, asfurther described supra.

Thus, the irradiance statistics for a specific satellite pixel, that is,an area statistic, rather than a point statistics, can be converted intothe irradiance statistics at an average point within that pixel bydividing by a “Area” term (A), which corresponds to the area of thesatellite pixel. Furthermore, the probability density function andcorrelation coefficient functions are generally assumed to be the samefor all pixels within the fleet region, making the value of A constantfor all pixels and reducing the computational burden further. Details asto how to calculate A are also further described supra.

$\begin{matrix}{\sigma_{Kt}^{2} = \frac{\sigma_{{Kt} - {Area}}^{2}}{A_{Kt}^{{Satellite}\;{Pixel}}}} & (7)\end{matrix}$

where:

$\begin{matrix}{A_{Kt}^{{Satellite}\mspace{11mu}{Pixel}} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (8)\end{matrix}$

Likewise, the change in clearness index variance across the satelliteregion can also be converted to an average point estimate using asimilar conversion factor, A_(ΔKt) ^(Area).

$\begin{matrix}{\sigma_{\Delta\;{Kt}}^{2} = \frac{\sigma_{{\Delta\;{Kt}} - {Area}}^{2}}{A_{\Delta\;{Kt}}^{SatellitePixel}}} & (9)\end{matrix}$

Variance of Clearness Index

$\sigma_{\overset{\_}{Kt}}^{2}$and Variance of Change in Clearness Index

$\sigma_{\overset{\_}{\Delta\;{Kt}}}^{2}$

At this point, the point statistics (σ_(Kt) ² and σ_(ΔKt) ²) have beendetermined for each of several representative locations within the fleetregion. These values may have been obtained from either ground-basedpoint data or by converting satellite data from area into pointstatistics. If the fleet region is small, the variances calculated ateach location i can be averaged to determine the average point varianceacross the fleet region. If there are N locations, then average varianceof the clearness index across the photovoltaic fleet region iscalculated as follows.

$\begin{matrix}{\sigma_{\overset{\_}{Kt}}^{2} = {\sum\limits_{i = 1}^{N}\;\frac{\sigma_{\;{Kt}_{i}}^{2}}{N}}} & (10)\end{matrix}$

Likewise, the variance of the clearness index change is calculated asfollows.

$\begin{matrix}{\sigma_{\overset{\_}{\Delta\;{Kt}}}^{2} = {\sum\limits_{i = 1}^{N}\;\frac{\sigma_{\;{\Delta\;{Kt}_{i}}}^{2}}{N}}} & (11)\end{matrix}$Calculate Fleet Power Statistics

The next step is to calculate photovoltaic fleet power statistics usingthe fleet irradiance statistics, as determined supra, and physicalphotovoltaic fleet configuration data. These fleet power statistics arederived from the irradiance statistics and have the same time period.

The critical photovoltaic fleet performance statistics that are ofinterest are the mean fleet power, the variance of the fleet power, andthe variance of the change in fleet power over the desired time period.As in the case of irradiance statistics, the mean change in fleet poweris assumed to be zero.

Photovoltaic System Power for Single System at Time t

Photovoltaic system power output (kW) is approximately linearly relatedto the AC-rating of the photovoltaic system (R in units of kW_(AC))times plane-of-array irradiance. Plane-of-array irradiance can berepresented by the clearness index over the photovoltaic system (KtPV)times the clear sky global horizontal irradiance times an orientationfactor (O), which both converts global horizontal irradiance toplane-of-array irradiance and has an embedded factor that convertsirradiance from Watts/m² to kW output/kW of rating. Thus, at a specificpoint in time (t), the power output for a single photovoltaic system (n)equals:P _(t) ^(n) =R ^(n) O _(t) ^(n) KtPV _(t) ^(n) I _(t) ^(Clear,n)  (12)

The change in power equals the difference in power at two differentpoints in time.ΔP _(t,Δt) ^(n) =R ^(n) O _(t+Δt) ^(n) KtPV _(t+Δt) ^(n) I _(t+Δt)^(Clear,n) −R ^(n) O _(t) ^(n) KtPV _(t) ^(n) I _(t) ^(Clear,n)  (13)

The rating is constant, and over a short time interval, the two clearsky plane-of-array irradiances are approximately the same (O_(t+Δt)^(n)I_(t+Δt) ^(Clear,n)≈O_(t) ^(n)I_(t) ^(Clear,n)), so that the threeterms can be factored out and the change in the clearness index remains.ΔP _(t,Δt) ^(n) ≈R ^(n) O _(t) ^(n) I _(t) ^(Clear,n) ΔKtPV _(t)^(n)  (14)

Time Series Photovoltaic Power for Single System

P^(n) is a random variable that summarizes the power for a singlephotovoltaic system n over a set of times for a given time interval andset of time periods. ΔP^(n) is a random variable that summarizes thechange in power over the same set of times.

Mean Fleet Power (μ_(P))

The mean power for the fleet of photovoltaic systems over the timeperiod equals the expected value of the sum of the power output from allof the photovoltaic systems in the fleet.

$\begin{matrix}{\mu_{p} = {E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{{KtPV}\;}^{n}I^{{Clear},n}}} \right\rbrack}} & (15)\end{matrix}$

If the time period is short and the region small, the clear skyirradiance does not change much and can be factored out of theexpectation.

$\begin{matrix}{\mu_{P} = {\mu_{I^{Clear}}{E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{\; n}}} \right\rbrack}}} & (16)\end{matrix}$

Again, if the time period is short and the region small, the clearnessindex can be averaged across the photovoltaic fleet region and any givenorientation factor can be assumed to be a constant within the timeperiod. The result is that:μ_(P) =R ^(Adj.Fleet)μ_(I) _(Clear) μKt   (17)

where μ_(I) _(Clear) is calculated, μKt taken from Equation (1) and:

$\begin{matrix}{R^{{Adj}.{Fleet}} = {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}}} & (18)\end{matrix}$

This value can also be expressed as the average power during clear skyconditions times the average clearness index across the region.μ_(P)=μ_(P) _(Clear) μ _(Kt)   (19)

Variance of Fleet Power (σ_(P) ²)

The variance of the power from the photovoltaic fleet equals:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{\; n}I^{{Clear},n}}} \right\rbrack}} & (20)\end{matrix}$

If the clear sky irradiance is the same for all systems, which will bethe case when the region is small and the time period is short, then:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {I^{Clear}{\sum\limits_{n = 1}^{N}{R^{n}O^{n}{{KtPV}\;}^{n}}}} \right\rbrack}} & (21)\end{matrix}$

The variance of a product of two independent random variables X, Y, thatis, VAR[XY]) equals E[X]²VAR[Y]+E[Y]²VAR[X]+VAR[X]VAR[Y]. If the Xrandom variable has a large mean and small variance relative to theother terms, then VAR[XY]≈E[X]²VAR[Y]. Thus, the clear sky irradiancecan be factored out of Equation (21) and can be written as:

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}{{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}{{KtPV}\;}^{n}O^{n}}} \right\rbrack}}} & (22)\end{matrix}$

The variance of a sum equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {{R^{i}{KtPV}^{i}O^{i}},{R^{j}{KtPV}^{j}O^{j}}} \right\rbrack}}} \right)}} & (23)\end{matrix}$

In addition, over a short time period, the factor to convert from clearsky GHI to clear sky POA does not vary much and becomes a constant. Allfour variables can be factored out of the covariance equation.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right){{COV}\left\lbrack {{KtPV}^{i},{KtPV}^{j}} \right\rbrack}}}} \right)}} & (24)\end{matrix}$

For any i and j, COV[KtPV^(i),KtPV^(j)]=√{square root over (σ_(KtPV)_(i) ²σ_(KTPV) _(j) ²)}ρ^(Kt) ^(i) ^(,Kt) ^(j) )

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right)\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}}\rho^{{Kt}^{i},{Kt}^{j}}}}} \right)}} & (25)\end{matrix}$

As discussed supra, the variance of the satellite data required aconversion from the satellite area, that is, the area covered by apixel, to an average point within the satellite area. In the same way,assuming a uniform clearness index across the region of the photovoltaicplant, the variance of the clearness index across a region the size ofthe photovoltaic plant within the fleet also needs to be adjusted. Thesame approach that was used to adjust the satellite clearness index canbe used to adjust the photovoltaic clearness index. Thus, each varianceneeds to be adjusted to reflect the area that the i^(th) photovoltaicplant covers.

$\begin{matrix}{\sigma_{{KtPV}^{i}}^{2} = {A_{Kt}^{i}\sigma\frac{2}{Kt}}} & (26)\end{matrix}$

Substituting and then factoring the clearness index variance given theassumption that the average variance is constant across the regionyields:

$\begin{matrix}{\sigma_{P}^{2} = {\left( {R^{{Adj}.{Fleet}}\mu_{I^{Clear}}} \right)^{2}P^{Kt}\sigma\frac{2}{Kt}}} & (27)\end{matrix}$

where the correlation matrix equals:

$\begin{matrix}{P^{Kt} = \frac{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}A_{Kt}^{i}} \right)\left( {R^{j}O^{j}A_{Kt}^{j}} \right)\rho^{{Kt}^{i},{Kt}^{j}}}}}{\left( {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (28)\end{matrix}$

R^(Adj.Fleet)μ_(I) _(Clear) in Equation (27) can be written as the powerproduced by the photovoltaic fleet under clear sky conditions, that is:

$\begin{matrix}{\sigma_{P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{Kt}\sigma_{\overset{\_}{Kt}}^{2}}} & (29)\end{matrix}$

If the region is large and the clearness index mean or variances varysubstantially across the region, then the simplifications may not beable to be applied. Notwithstanding, if the simplification isinapplicable, the systems are likely located far enough away from eachother, so as to be independent. In that case, the correlationcoefficients between plants in different regions would be zero, so mostof the terms in the summation are also zero and an inter-regionalsimplification can be made. The variance and mean then become theweighted average values based on regional photovoltaic capacity andorientation.

Discussion

In Equation (28), the correlation matrix term embeds the effect ofintra-plant and inter-plant geographic diversification. The area-relatedterms (A) inside the summations reflect the intra-plant power smoothingthat takes place in a large plant and may be calculated using thesimplified relationship, as further discussed supra. These terms arethen weighted by the effective plant output at the time, that is, therating adjusted for orientation. The multiplication of these terms withthe correlation coefficients reflects the inter-plant smoothing due tothe separation of photovoltaic systems from one another.

Variance of Change in Fleet Power (σ_(ΔP) ²)

A similar approach can be used to show that the variance of the changein power equals:

$\begin{matrix}{\sigma_{\Delta\; P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{\Delta\;{Kt}}\sigma_{\overset{\_}{\Delta\;{Kt}}}^{2}}} & (30) \\{P^{\Delta\;{Kt}} = \frac{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}A_{\Delta\;{Kt}}^{i}} \right)\left( {R^{j}O^{j}A_{\Delta\;{Kt}}^{j}} \right)\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{j}}}}}}{\left( {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (31)\end{matrix}$

The determination of Equations (30) and (31) becomes computationallyintensive as the network of points becomes large. For example, a networkwith 10,000 photovoltaic systems would require the computation of acorrelation coefficient matrix with 100 million calculations. Thecomputational burden can be reduced in two ways. First, many of theterms in the matrix are zero because the photovoltaic systems arelocated too far away from each other. Thus, the double summation portionof the calculation can be simplified to eliminate zero values based ondistance between locations by construction of a grid of points. Second,once the simplification has been made, rather than calculating thematrix on-the-fly for every time period, the matrix can be calculatedonce at the beginning of the analysis for a variety of cloud speedconditions, and then the analysis would simply require a lookup of theappropriate value.

Time Lag Correlation Coefficient

The next step is to adjust the photovoltaic fleet power statistics fromthe input time interval to the desired output time interval. Forexample, the time series data may have been collected and stored every60 seconds. The user of the results, however, may want to havephotovoltaic fleet power statistics at a 10-second rate. This adjustmentis made using the time lag correlation coefficient.

The time lag correlation coefficient reflects the relationship betweenfleet power and that same fleet power starting one time interval (Δt)later. Specifically, the time lag correlation coefficient is defined asfollows:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = \frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}} & (32)\end{matrix}$

The assumption that the mean clearness index change equals zero impliesthat σ_(P) _(Δt) ²=σ_(P) ². Given a non-zero variance of power, thisassumption can also be used to show that

$\begin{matrix}{\frac{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack}{\sigma_{P}^{2}} = {1 - {\frac{\sigma_{\;^{\Delta\; P}}^{2}}{2\sigma_{P}^{2}}.}}} & \;\end{matrix}$Therefore:

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{\sigma_{\;^{\Delta\; P}}^{2}}{2\sigma_{P}^{2}}}} & (33)\end{matrix}$

This relationship illustrates how the time lag correlation coefficientfor the time interval associated with the data collection rate iscompletely defined in terms of fleet power statistics alreadycalculated. A more detailed derivation is described infra.

Equation (33) can be stated completely in terms of the photovoltaicfleet configuration and the fleet region clearness index statistics bysubstituting Equations (29) and (30). Specifically, the time lagcorrelation coefficient can be stated entirely in terms of photovoltaicfleet configuration, the variance of the clearness index, and thevariance of the change in the clearness index associated with the timeincrement of the input data.

$\begin{matrix}{\rho^{P,P^{\Delta\; t}} = {1 - \frac{P^{\Delta\;{Kt}}\sigma_{\overset{\_}{\Delta\;{Kt}}}^{2}}{2P^{Kt}\sigma_{\overset{\_}{\;{Kt}}}^{2}}}} & (34)\end{matrix}$Generate High-Speed Time Series Photovoltaic Fleet Power

The final step is to generate high-speed time series photovoltaic fleetpower data based on irradiance statistics, photovoltaic fleetconfiguration, and the time lag correlation coefficient. This step is toconstruct time series photovoltaic fleet production from statisticalmeasures over the desired time period, for instance, at half-hour outputintervals.

A joint probability distribution function is required for this step. Thebivariate probability density function of two unit normal randomvariables (X and Y) with a correlation coefficient of ρ equals:

$\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{2\pi\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {x^{2} + y^{2} - {2\;\rho\;{xy}}} \right)}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (35)\end{matrix}$

The single variable probability density function for a unit normalrandom variable X alone is

${f(x)} = {\frac{1}{\sqrt{2\pi}}{{\exp\left( {- \frac{x^{2}}{2}} \right)}.}}$In addition, a conditional distribution for y can be calculated based ona known x by dividing the bivariate probability density function by thesingle variable probability density (i.e., f(y|x)=f(x,y)/f(x)). Makingthe appropriate substitutions, the result is that the conditionaldistribution of y based on a known x equals:

$\begin{matrix}{{f\left( y \middle| x \right)} = {\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^{2}}}{\exp\left\lbrack {- \frac{\left( {y - {px}} \right)^{2}}{2\left( {1 - p^{2}} \right)}} \right\rbrack}}} & (36)\end{matrix}$

Define a random variable

$Z = \frac{Y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}$and substitute into Equation (36). The result is that the conditionalprobability of z given a known x equals:

$\begin{matrix}{{f\left( z \middle| x \right)} = {\frac{1}{\sqrt{2\pi}}{\exp\left( {- \frac{z^{2}}{2}} \right)}}} & (37)\end{matrix}$

The cumulative distribution function for Z can be denoted by φ(z*),where z* represents a specific value for z. The result equals aprobability (p) that ranges between 0 (when z*=−∞) and 1 (when z*=∞).The function represents the cumulative probability that any value of zis less than z*, as determined by a computer program or value lookup.

$\begin{matrix}{p = {{\phi\left( z^{*} \right)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{z^{*}}{{\exp\left( {- \frac{z^{2}}{2}} \right)}{\mathbb{d}z}\int}}}}} & (38)\end{matrix}$

Rather than selecting z*, however, a probability p falling between 0 and1 can be selected and the corresponding z* that results in thisprobability found, which can be accomplished by taking the inverse ofthe cumulative distribution function.φ⁻¹(p)=z*  (39)

Substituting back for z as defined above results in:

$\begin{matrix}{{\phi^{- 1}(p)} = \frac{y - {\rho\; x}}{\sqrt{1 - \rho^{2}}}} & (40)\end{matrix}$

Now, let the random variables equal

$X = \frac{P - \mu_{P}}{\sigma_{P}}$and

${Y = \frac{P^{\Delta\; t} - \mu_{P^{\Delta\; t}}}{\sigma_{P^{\Delta\; t}}}},$with the correlation coefficient being the time lag correlationcoefficient between P and P^(Δt) (i.e., let μ=μ^(P,P) ^(Δt) ). When Δtis small, then the mean and standard deviations for P^(Δt) areapproximately equal to the mean and standard deviation for P. Thus, Ycan be restated as

$Y \approx {\frac{P^{\Delta\; t} - \mu_{P}}{\sigma_{P}}.}$Add a time subscript to all of the relevant data to represent a specificpoint in time and substitute x, y, and ρ into Equation (40).

$\begin{matrix}{{\phi^{- 1}(p)} = \frac{\left( \frac{{P^{\Delta\; t}t} - \mu_{P}}{\sigma_{P}} \right) - {\rho^{P,P^{\Delta\; t}}\left( \frac{P_{t} - \mu_{P}}{\sigma_{P}} \right)}}{\sqrt{1 - \rho^{P,P^{\Delta\; t^{2}}}}}} & (41)\end{matrix}$

The random variable P^(Δt), however, is simply the random variable Pshifted in time by a time interval of Δt. As a result, at any given timet, P^(Δt) _(t)=P_(t+Δt). Make this substitution into Equation (41) andsolve in terms of P_(t+Δt).

$\begin{matrix}{P_{t + {\Delta\; t}} = {{\rho^{P,P^{\Delta\; t}}P_{t}} + {\left( {1 - \rho^{P,P^{\Delta\; t}}} \right)\mu_{P}} + {\sqrt{\sigma_{P}^{2}\left( {1 - \rho^{P,P^{\Delta\; t^{2}}}} \right)}{\phi^{- 1}(p)}}}} & (42)\end{matrix}$

At any given time, photovoltaic fleet power equals photovoltaic fleetpower under clear sky conditions times the average regional clearnessindex, that is, P_(t)=P_(t) ^(Clear)Kt_(t). In addition, over a shorttime period, μ_(p)≈P_(t) ^(Clear)μ _(Kt) and

$\sigma_{P}^{2} \approx {\left( P_{t}^{Clear} \right)^{2}P^{Kt}\sigma{\frac{2}{Kt}.}}$Substitute these three relationships into Equation (42) and factor outphotovoltaic fleet power under clear sky conditions ((P_(t) ^(Clear)))as common to all three terms.

$\begin{matrix}{P_{t + {\Delta\; t}} = {P_{t}^{Clear}\left\lbrack {{\rho^{P,P^{\Delta\; t}}{Kt}_{t}} + {{\quad\quad}\left( {1 - \rho^{P,P^{\Delta\; t}}} \right)\mu_{\overset{\_}{Kt}}} + {\sqrt{P^{Kt}\sigma\frac{2}{Kt}\left( {1 - \rho^{P,P^{\Delta\; t^{2}}}} \right)}{\phi^{- 1}\left( p_{t} \right)}}} \right\rbrack}} & (43)\end{matrix}$

Equation (43) provides an iterative method to generate high-speed timeseries photovoltaic production data for a fleet of photovoltaic systems.At each time step (t+Δt), the power delivered by the fleet ofphotovoltaic systems (P_(t+Δt)) is calculated using input values fromtime step t. Thus, a time series of power outputs can be created. Theinputs include:

-   -   P_(t) ^(Clear)—photovoltaic fleet power during clear sky        conditions calculated using a photovoltaic simulation program        and clear sky irradiance.,    -   Kt_(t)—average regional clearness index inferred based on P_(t)        calculated in time step t, that is,

${Kt}_{t} = {\frac{P_{t}}{P_{t}^{Clear}}.}$

-   -   μ _(Kt) —mean clearness index calculated using time series        irradiance data and Equation (1).

$\sigma\frac{2}{Kt}$

-   -    —variance of the clearness index calculated using time series        irradiance data and Equation (10).    -   ρ^(P,P) ^(Δt) —fleet configuration as reflected in the time lag        correlation coefficient calculated using Equation (34). In turn,        Equation (34), relies upon correlation coefficients from        Equations (28) and (31). A method to obtain these correlation        coefficients by empirical means is described herein.    -   P^(Kt)—fleet configuration as reflected in the clearness index        correlation coefficient matrix calculated using Equation (28)        where, again, the correlation coefficients may be obtained using        the empirical results as further described infra.    -   φ⁻¹(p_(t))—the inverse cumulative normal distribution function        based on a random variable between 0 and 1.        Derivation of Empirical Models

The previous section developed the mathematical relationships used tocalculate irradiance and power statistics for the region associated witha photovoltaic fleet. The relationships between Equations (8), (28),(31), and (34) depend upon the ability to obtain point-to-pointcorrelation coefficients. This section presents empirically-derivedmodels that can be used to determine the value of the coefficients forthis purpose.

A mobile network of 25 weather monitoring devices was deployed in a 400meter by 400 meter grid in Cordelia Junction, Calif., between Nov. 6,2010, and Nov. 15, 2010, and in a 4,000 meter by 4,000 meter grid inNapa, Calif., between Nov. 19, 2010, and Nov. 24, 2010. FIGS. 7A-7B arephotographs showing, by way of example, the locations of the CordeliaJunction and Napa high density weather monitoring stations.

An analysis was performed by examining results from Napa and CordeliaJunction using 10, 30, 60, 120 and 180 second time intervals over eachhalf-hour time period in the data set. The variance of the clearnessindex and the variance of the change in clearness index were calculatedfor each of the 25 locations for each of the two networks. In addition,the clearness index correlation coefficient and the change in clearnessindex correlation coefficient for each of the 625 possible pairs, 300 ofwhich are unique, for each of the two locations were calculated.

An empirical model is proposed as part of the methodology describedherein to estimate the correlation coefficient of the clearness indexand change in clearness index between any two points by using as inputsthe following: distance between the two points, cloud speed, and timeinterval. For the analysis, distances were measured, cloud speed wasimplied, and a time interval was selected.

The empirical models infra describe correlation coefficients between twopoints (i and j), making use of “temporal distance,” defined as thephysical distance (meters) between points i and j, divided by theregional cloud speed (meters per second) and having units of seconds.The temporal distance answers the question, “How much time is needed tospan two locations?”

Cloud speed was estimated to be six meters per second. Results indicatethat the clearness index correlation coefficient between the twolocations closely matches the estimated value as calculated using thefollowing empirical model:ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C₁×TemporalDistance)^(ClearnessPower)  (44)

where TemporalDistance=Distance (meters)/CloudSpeed (meters per second),Clearness Power=ln(C₂Δt)−k, such that 5≦k≦15, where the expected valueis k=9.3, Δt is the desired output time interval (seconds), and C₁=10⁻³seconds⁻¹, and C₂=1 seconds⁻¹.

Results also indicate that the correlation coefficient for the change inclearness index between two locations closely matches the valuescalculated using the following empirical relationship:ρ^(Kt) ^(i) ^(,ΔKt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j))^(ΔClearnessPower)  (45)

where ρ^(Kt) ^(i) ^(,Kt) ^(j) is calculated using Equation (44) and

${{\Delta\;{ClearnessPower}} = {1 + \frac{m}{C_{2}\Delta\; t}}},$such that 100≦m≦200, where the expected value is m=140.

Empirical results also lead to the following models that may be used totranslate the variance of clearness index and the variance of change inclearness index from the measured time interval (Δt ref) to the desiredoutput time interval (Δt).

$\begin{matrix}{\sigma_{{Kt}_{\Delta\; t}}^{2} = {\sigma_{{Kt}_{\Delta\; t_{ref}}}^{2}{\exp\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{3}}} \right\rbrack}}} & (46) \\{\sigma_{\Delta\;{Kt}_{\Delta\; t}}^{2} = {\sigma_{\Delta\;{Kt}_{\Delta\;{tref}}}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta\; t}{\Delta\; t\mspace{14mu}{ref}} \right)^{C_{3}}} \right\rbrack}} \right\}}} & (47)\end{matrix}$

where C₃=0.1≦C₃≦0.2, where the expected value is C₃=0.15.

FIGS. 8A-8B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds. Forexample, if the variance is calculated at a 300-second time interval andthe user desires results at a 10-second time interval, the adjustmentfor the variance clearness index would be 1.49

These empirical models represent a valuable means to rapidly calculatecorrelation coefficients and translate time interval withreadily-available information, which avoids the use ofcomputation-intensive calculations and high-speed streams of data frommany point sources, as would otherwise be required.

Validation

Equations (44) and (45) were validated by calculating the correlationcoefficients for every pair of locations in the Cordelia Junctionnetwork and the Napa network at half-hour time periods. The correlationcoefficients for each time period were then weighted by thecorresponding variance of that location and time period to determineweighted average correlation coefficient for each location pair. Theweighting was performed as follows:

${\overset{\_}{\rho^{{Kt}^{i},{Kt}^{j}}} = \frac{\sum\limits_{t = 1}^{T}{\sigma_{{{Kt} - i},j_{t}}^{2}\rho^{{Kt}^{i},{Kt}_{t}^{j}}}}{\sum\limits_{t = 1}^{T}\sigma_{{{Kt} - i},j_{t}}^{2}}},\mspace{14mu}{and}$$\overset{\_}{\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}^{j}}}} = {\frac{\sum\limits_{t = 1}^{T}{\sigma_{{{\Delta\;{Kt}} - i},j_{t}}^{2}\rho^{{\Delta\;{Kt}^{i}},{\Delta\;{Kt}_{t}^{j}}}}}{\sum\limits_{i = 1}^{T}\sigma_{{{\Delta\;{Kt}} - i},j_{t}}^{2}}.}$

FIGS. 9A-9F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance. FIGS. 10A-10F are graphs depicting, by way ofexample, the same information as depicted in FIGS. 9A-9F versus temporaldistance, based on the assumption that cloud speed was 6 meters persecond. The upper line and dots appearing in close proximity to theupper line present the clearness index and the lower line and dotsappearing in close proximity to the lower line present the change inclearness index for time intervals from 10 seconds to 5 minutes. Thesymbols are the measured results and the lines are the predictedresults.

Several observations can be drawn based on the information provided bythe FIGS. 9A-9F and 10A-10F. First, for a given time interval, thecorrelation coefficients for both the clearness index and the change inthe clearness index follow an exponential decline pattern versusdistance (and temporal distance). Second, the predicted results are agood representation of the measured results for both the correlationcoefficients and the variances, even though the results are for twoseparate networks that vary in size by a factor of 100. Third, thechange in the clearness index correlation coefficient converges to theclearness correlation coefficient as the time interval increases. Thisconvergence is predicted based on the form of the empirical modelbecause ΔClearness Power approaches one as Δt becomes large.

Equation (46) and (47) were validated by calculating the averagevariance of the clearness index and the variance of the change in theclearness index across the 25 locations in each network for everyhalf-hour time period. FIGS. 11A-11F are graphs depicting, by way ofexample, the predicted versus the measured variances of clearnessindexes using different reference time intervals. FIGS. 12A-12F aregraphs depicting, by way of example, the predicted versus the measuredvariances of change in clearness indexes using different reference timeintervals. FIGS. 11A-11F and 12A-12F suggest that the predicted resultsare similar to the measured results.

Discussion

The point-to-point correlation coefficients calculated using theempirical forms described supra refer to the locations of specificphotovoltaic power production sites. Importantly, note that the dataused to calculate these coefficients was not obtained from time sequencemeasurements taken at the points themselves. Rather, the coefficientswere calculated from fleet-level data (cloud speed), fixed fleet data(distances between points), and user-specified data (time interval).

The empirical relationships of the foregoing types of empiricalrelationships may be used to rapidly compute the coefficients that arethen used in the fundamental mathematical relationships. The methodologydoes not require that these specific empirical models be used andimproved models will become available in the future with additional dataand analysis.

Example

This section provides a complete illustration of how to apply themethodology using data from the Napa network of 25 irradiance sensors onNov. 21, 2010. In this example, the sensors served as proxies for anactual 1 kW photovoltaic fleet spread evenly over the geographicalregion as defined by the sensors. For comparison purposes, a directmeasurement approach is used to determine the power of this fleet andthe change in power, which is accomplished by adding up the 10-secondoutput from each of the sensors and normalizing the output to a 1 kWsystem. FIGS. 13A-13F are graphs and a diagram depicting, by way ofexample, application of the methodology described herein to the Napanetwork.

The predicted behavior of the hypothetical photovoltaic fleet wasseparately estimated using the steps of the methodology described supra.The irradiance data was measured using ground-based sensors, althoughother sources of data could be used, including from existingphotovoltaic systems or satellite imagery. As shown in FIG. 13A, thedata was collected on a day with highly variable clouds with one-minuteglobal horizontal irradiance data collected at one of the 25 locationsfor the Napa network and specific 10-second measured power outputrepresented by a blue line. This irradiance data was then converted fromglobal horizontal irradiance to a clearness index. The mean clearnessindex, variance of clearness index, and variance of the change inclearness index was then calculated for every 15-minute period in theday. These calculations were performed for each of the 25 locations inthe network. Satellite-based data or a statistically-significant subsetof the ground measurement locations could have also served in place ofthe ground-based irradiance data. However, if the data had beencollected from satellite regions, an additional translation from areastatistics to average point statistics would have been required. Theaveraged irradiance statistics from Equations (1), (10), and (11) areshown in FIG. 13B, where standard deviation (σ) is presented, instead ofvariance (σ²) to plot each of these values in the same units.

In this example, the irradiance statistics need to be translated sincethe data were recorded at a time interval of 60 seconds, but the desiredresults are at a 10-second resolution. The translation was performedusing Equations (46) and (47) and the result is presented in FIG. 13C.

The details of the photovoltaic fleet configuration were then obtained.The layout of the fleet is presented in FIG. 13D. The details includethe location of the each photovoltaic system (latitude and longitude),photovoltaic system rating ( 1/25 kW), and system orientation (all arehorizontal).

Equation (43), and its associated component equations, were used togenerate the time series data for the photovoltaic fleet with theadditional specification of the specific empirical models, as describedin Equations (44) through (47). The resulting fleet power and change inpower is presented represented by the red lines in FIGS. 12E and 12F.

Probability Density Function

The conversion from area statistics to point statistics relied upon twoterms A_(Kt) and A_(ΔKt) to calculate σ_(Kt) ² and σ_(ΔKt) ²,respectively. This section considers these terms in more detail. Forsimplicity, the methodology supra applies to both Kt and ΔKt, so thisnotation is dropped. Understand that the correlation coefficient ρ^(i,j)could refer to either the correlation coefficient for clearness index orthe correlation coefficient for the change in clearness index, dependingupon context. Thus, the problem at hand is to evaluate the followingrelationship:

$\begin{matrix}{A = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (48)\end{matrix}$

The computational effort required to calculate the correlationcoefficient matrix can be substantial. For example, suppose that the onewants to evaluate variance of the sum of points within a 1 squarekilometer satellite region by breaking the region into one millionsquare meters (1,000 meters by 1,000 meters). The complete calculationof this matrix requires the examination of 1 trillion (10¹²) locationpair combinations.

Discrete Formulation

The calculation can be simplified using the observation that many of theterms in the correlation coefficient matrix are identical. For example,the covariance between any of the one million points and themselvesis 1. This observation can be used to show that, in the case of arectangular region that has dimension of H by W points (total of N) andthe capacity is equal distributed across all parts of the region that:

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\begin{bmatrix}{{2^{k}\left\lbrack {\left( {H - i} \right)\left( {W - j} \right)} \right\rbrack}\rho^{d}{\sum\limits_{i = 0}^{H - 1}{\sum\limits_{j = 0}^{i} +}}} \\{\sum\limits_{i = 0}^{W - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {W - i} \right)\left( {H - j} \right)} \right\rbrack}\rho^{d}}}}\end{bmatrix}}} & (49)\end{matrix}$

where:

-   -   −1, when i=0 and j=0

k=1, when j=0 or j=i

-   -   2, when 0<j<i

When the region is a square, a further simplification can be made.

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = 0}^{i}{2^{k}\left( {\sqrt{N} - i} \right)\left( {\sqrt{N} - j} \right)\rho^{d}}}} \right\rbrack}} & (50)\end{matrix}$

where:

-   -   0, when i=0 and j=0

k=2, when j=0 or j=i, and

-   -   3, when 0<j<i

$d = {\left( \sqrt{i^{2} + j^{2}} \right){\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right).}}$

The benefit of Equation (50) is that there are

$\frac{N - \sqrt{N}}{2}$rather than N² unique combinations that need to be evaluated. In theexample above, rather than requiring one trillion possible combinations,the calculation is reduced to one-half million possible combinations.

Continuous Formulation

Even given this simplification, however, the problem is stillcomputationally daunting, especially if the computation needs to beperformed repeatedly in the time series. Therefore, the problem can berestated as a continuous formulation in which case a proposedcorrelation function may be used to simplify the calculation. The onlyvariable that changes in the correlation coefficient between any of thelocation pairs is the distance between the two locations; all othervariables are the same for a given calculation. As a result, Equation(50) can be interpreted as the combination of two factors: theprobability density function for a given distance occurring and thecorrelation coefficient at the specific distance.

Consider the probability density function. The actual probability of agiven distance between two pairs occurring was calculated for a 1,000meter×1,000 meter grid in one square meter increments. The evaluation ofone trillion location pair combination possibilities was evaluated usingEquation (48) and by eliminating the correlation coefficient from theequation. FIG. 14 is a graph depicting, by way of example, an actualprobability distribution for a given distance between two pairs oflocations, as calculated for a 1,000 meter×1,000 meter grid in onesquare meter increments.

The probability distribution suggests that a continuous approach can betaken, where the goal is to find the probability density function basedon the distance, such that the integral of the probability densityfunction times the correlation coefficient function equals:A=∫f(D)ρ(d)dD  (51)

An analysis of the shape of the curve shown in FIG. 14 suggests that thedistribution can be approximated through the use of two probabilitydensity functions. The first probability density function is a quadraticfunction that is valid between 0 and √{square root over (Area)}.

$\begin{matrix}{f_{Quad} = \left\{ \begin{matrix}{\left( \frac{6}{Area} \right)\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)} & {{{for}\mspace{14mu} 0} \leq D \leq \sqrt{Area}} \\0 & {{{for}\mspace{14mu} D} > \sqrt{Area}}\end{matrix} \right.} & (52)\end{matrix}$

This function is a probability density function because integratingbetween

$0\mspace{14mu}{and}\mspace{14mu}\sqrt{Area}\mspace{14mu}{equals}\mspace{14mu} 1{\left( {{i.e.},{{P\left\lbrack {0 \leq D \leq \sqrt{Area}} \right\rbrack} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\mathbb{d}D}}} = 1}}} \right).}$

The second function is a normal distribution with a mean of √{squareroot over (Area)} and standard deviation of 0.1√{square root over(Area)}.

$\begin{matrix}{f_{Norm} = {\left( \frac{1}{0.1*\sqrt{Area}} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\mathbb{e}}^{{- {(\frac{1}{2})}}{(\frac{D - \sqrt{Area}}{0.1*\sqrt{Area}})}^{2}}}} & (53)\end{matrix}$Likewise, integrating across all values equals 1.

To construct the desired probability density function, take, forinstance, 94 percent of the quadratic density function plus 6 of thenormal density function.

$\begin{matrix}{f = {{0.94{\int_{0}^{\sqrt{Area}}{f_{Quad}{\mathbb{d}D}}}} + {0.06{\int_{- \infty}^{+ \infty}{f_{Norm}{\mathbb{d}D}}}}}} & (54)\end{matrix}$

FIG. 15 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

The result is that the correlation matrix of a square area with uniformpoint distribution as N gets large can be expressed as follows, firstdropping the subscript on the variance since this equation will work forboth Kt and ΔKt.

$\begin{matrix}{A \approx \left\lbrack {{0.94{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho(D)}{\mathbb{d}D}}}} + {0.06{\int_{- \infty}^{+ \infty}{f_{Norm}{\rho(D)}{\mathbb{d}D}}}}} \right\rbrack} & (55)\end{matrix}$

where ρ(D) is a function that expresses the correlation coefficient as afunction of distance (D).

Area to Point Conversion Using Exponential Correlation Coefficient

Equation (55) simplifies the problem of calculating the correlationcoefficient and can be implemented numerically once the correlationcoefficient function is known. This section demonstrates how a closedform solution can be provided, if the functional form of the correlationcoefficient function is exponential.

Noting the empirical results as shown in the graph in FIGS. 9A-9F, anexponentially decaying function can be taken as a suitable form for thecorrelation coefficient function. Assume that the functional form ofcorrelation coefficient function equals:

$\begin{matrix}{{\rho(D)} = {\mathbb{e}}^{\frac{xD}{\sqrt{Area}}}} & (56)\end{matrix}$

Let Quad be the solution to

$\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho(D)}{{\mathbb{d}D}.}}$

$\begin{matrix}{{Quad} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho(D)}{\mathbb{d}D}}} = {\left( \frac{6}{Area} \right){\int_{0}^{\sqrt{Area}}{{\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)\left\lbrack {\mathbb{e}}^{\frac{xD}{\sqrt{Area}}} \right\rbrack}{\mathbb{d}D}}}}}} & (57)\end{matrix}$

Integrate to solve.

$\begin{matrix}{{Quad} = {(6)\left\lbrack {{\left( {{\frac{x}{\sqrt{Area}}D} - 1} \right){\mathbb{e}}^{\frac{xD}{\sqrt{Area}}}} - {\left( {{\left( \frac{x}{\sqrt{Area}} \right)^{2}D^{2}} - {2\frac{x}{\sqrt{Area}}D} + 2} \right){\mathbb{e}}^{\frac{xD}{\sqrt{Area}}}}} \right\rbrack}} & (58)\end{matrix}$

Complete the result by evaluating at D equal to √{square root over(Area)} for the upper bound and 0 for the lower bound. The result is:

$\begin{matrix}{{Quad} = {\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {{\mathbb{e}}^{x} + 1} \right)} + 4} \right\rbrack}} & (59)\end{matrix}$

Next, consider the solution to

∫_(−∞)^(+∞)f_(Norm)ρ^((D)𝕕D),which will be called Norm.

$\begin{matrix}{{Norm} = {\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{{\mathbb{e}}^{{- {(\frac{1}{2})}}{(\frac{D - \mu}{\sigma})}^{2}}{\mathbb{e}}^{\frac{xD}{\sqrt{Area}}}{\mathbb{d}D}}}}} & (60)\end{matrix}$

Where μ=√{square root over (Area)} and σ=0.1√{square root over (Area)}.Simplifying:

$\begin{matrix}\begin{matrix}{{Norm} = {\left\lbrack {\mathbb{e}}^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}} \right\rbrack{\left( \frac{1}{\sigma} \right) \cdot}}} \\{\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{{\mathbb{e}}^{- {{(\frac{1}{2})}\lbrack\frac{D - {({\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}})}}{\sigma}\rbrack}^{2}}{\mathbb{d}D}}}} \\{{{Substitute}\mspace{20mu} z} = {{\frac{D - \left( {\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}} \right)}{\sigma}\mspace{14mu}{and}\mspace{14mu}\sigma\;{dz}} = {{dD}.}}}\end{matrix} & (61) \\{{Norm} = {\left\lbrack {\mathbb{e}}^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}} \right\rbrack\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{{{\mathbb{e}}^{{- {(\frac{1}{2})}}z^{2}}{\mathbb{d}z}}{{Integrate}\mspace{14mu}{and}\mspace{14mu}{{solve}.}}}}}} & (62) \\{{Norm} = {\mathbb{e}}^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}}} & (63)\end{matrix}$

Substitute the mean of √{square root over (Area)} and the standarddeviation of 0.1√{square root over (Area)} into Equation (63).Norm=e ^(x(1+0.005) x)  (64)

Substitute the solutions for Quad and Norm back into Equation (55). Theresult is the ratio of the area variance to the average point variance.This ratio was referred to as A (with the appropriate subscripts andsuperscripts) supra.

$\begin{matrix}{A = {{0.94{\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {{\mathbb{e}}^{x} + 1} \right)} + 4} \right\rbrack}} + {0.06{\mathbb{e}}^{x{({1 + {0.005x}})}}}}} & (65)\end{matrix}$

Example

This section illustrates how to calculate A for the clearness index fora satellite pixel that covers a geographical surface area of 1 km by 1km (total area of 1,000,000 m²), using a 60-second time interval, and 6meter per second cloud speed. Equation (56) required that thecorrelation coefficient be of the form

${\mathbb{e}}^{\frac{xD}{\sqrt{Area}}}.$The empirically derived result in Equation (44) can be rearranged andthe appropriate substitutions made to show that the correlationcoefficient of the clearness index equals

${\exp\left\lbrack \frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)D}{1000{CloudSpeed}} \right\rbrack}.$Multiply the exponent by

$\frac{\sqrt{Area}}{\sqrt{Area}},$so that the correlation coefficient equals

$\exp{\left\{ {\left\lbrack \frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)\sqrt{Area}}{1000{CloudSpeed}} \right\rbrack\left\lbrack \frac{D}{\sqrt{Area}} \right\rbrack} \right\}.}$This expression is now in the correct form to apply Equation (65), where

$x = {\frac{\left( {{\ln\;\Delta\; t} - 9.3} \right)\sqrt{Area}}{1000{CloudSpeed}}.}$Inserting the assumptions results in

${x = {\frac{\left( {{\ln\; 60} - 9.3} \right)\sqrt{\text{1,000,000}}}{1000 \times \; 6} = {- 0.86761}}}\;,$which is applied to Equation (65). The result is that A equals 65percent, that is, the variance of the clearness index of the satellitedata collected over a 1 km² region corresponds to 65 percent of thevariance when measured at a specific point. A similar approach can beused to show that the A equals 27 percent for the change in clearnessindex. FIG. 16 is a graph depicting, by way of example, resultsgenerated by application of Equation (65).Time Lag Correlation Coefficient

This section presents an alternative approach to deriving the time lagcorrelation coefficient. The variance of the sum of the change in theclearness index equals:σ_(ΣΔKt) ² =VAR[E(Kt ^(Δt) −Kt)]  (66)where the summation is over N locations. This value and thecorresponding subscripts have been excluded for purposes of notationalsimplicity.

Divide the summation into two parts and add several constants to theequation:

$\begin{matrix}{\sigma_{\sum{\Delta\; K\; t}}^{2} = {{VAR}\left\lbrack {{\sigma_{\sum{Kt}^{\Delta\; t}}\left( \frac{\sum{Kt}^{\Delta\; t}}{\sigma_{\sum{Kt}^{\Delta\; t}}} \right)} - {\sigma_{\sum{Kt}}\left( \frac{\sum{Kt}}{\sigma_{\sum{Kt}}} \right)}} \right\rbrack}} & (67)\end{matrix}$

Since σ_(ΣKt) _(Δt) ≈σ_(ΣKt) (or σ_(ΣKt) _(Δt) =σ_(ΣKt) if the firstterm in Kt and the last term in Kt^(Δt) are the same):

$\begin{matrix}{\sigma_{\sum{\Delta\;{Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}{{VAR}\left\lbrack {\frac{\sum{Kt}^{\Delta\; t}}{\sigma_{\sum{Kt}^{\Delta\; t}}} - \frac{\sum{Kt}}{\sigma_{\sum{Kt}}}} \right\rbrack}}} & (68)\end{matrix}$

The variance term can be expanded as follows:

$\begin{matrix}{\sigma_{\sum{\Delta\;{Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}\left\{ {\frac{{VAR}\left\lbrack {\sum{Kt}^{\Delta\; t}} \right\rbrack}{\sigma_{\sum{Kt}^{\Delta\; t}}^{2}} + \frac{{VAR}\left\lbrack {\sum{Kt}} \right\rbrack}{\sigma_{\sum{Kt}}^{2}} - \frac{2{{COV}\left\lbrack {{\sum{Kt}},{\sum{Kt}^{\Delta\; t}}} \right\rbrack}}{\sigma_{\sum{Kt}}\sigma_{\sum{Kt}^{\Delta\; t}}}} \right\}}} & (69)\end{matrix}$

Since COV[ΣKt,ΣKt^(Δt)]=σ_(ΣKt)σ_(ΣKt) _(Δt) ρ^(Σkt,ΣKt) ^(Δt) , thefirst two terms equal one and the covariance term is replaced by thecorrelation coefficient.σ_(ΣΔKt) ²=2σ_(ΣKt) ²(1−ρ^(ΣKtmΣKt) ^(Δt) )  (70)

This expression rearranges to:

$\begin{matrix}{\rho^{\sum{Kt}},{{\sum{Kt}^{\Delta\; t}} = {1 - \left( {\frac{1}{2}\frac{\sigma_{\Delta\;{Kt}}^{2}}{\sigma_{Kt}^{2}}} \right)}}} & (71)\end{matrix}$

Assume that all photovoltaic plant ratings, orientations, and areaadjustments equal to one, calculate statistics for the clearness aloneusing the equations described supra and then substitute. The result is:

$\begin{matrix}{\rho^{{\sum{Kt}},{\sum{Kt}^{\Delta\; t}}} = {1 - \frac{P^{\Delta\;{Kt}}\sigma\frac{2}{\Delta\;{Kt}}}{2P^{Kt}\sigma\frac{2}{Kt}}}} & (72)\end{matrix}$Relationship Between Time Lag Correlation Coefficient and Power/Changein Power Correlation Coefficient

This section derives the relationship between the time lag correlationcoefficient and the correlation between the series and the change in theseries for a single location.

$\rho^{P,{\Delta\; P}} = {\frac{{COV}\left\lbrack {p,{\Delta\; P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = {\frac{{COV}\left\lbrack {P,{P^{\Delta\; t} - P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}} = \frac{{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta\; P}^{2}}}}}$Since  σ_(Δ P)² = VAR[P^(Δ t) − P] = σ_(P)² + σ_(P^(Δ t))² − 2COV[P, P^(Δ t)], and${{{COV}\left\lbrack {P,P^{\Delta\; t}} \right\rbrack} = {\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}},{{{then}:\rho^{P,{\Delta\; P}}} = \frac{{\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\left( {\sigma_{P}^{2} + \sigma_{P^{\Delta\; t}}^{2} - {2\rho^{P,P^{\Delta\; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta\; t}}^{2}}}} \right)}}}$

Since σ_(P) ²≈σ_(P) _(Δt) ², this expression can be further simplified.Then, square both expression and solve for the time lag correlationcoefficient:ρ^(P,P) ^(Δt) =1−2(ρ^(P,ΔP))²Correlation Coefficients Between Two Regions

Assume that the two regions are squares of the same size, each side withN points, that is, a matrix with dimensions of √{square root over (N)}by √{square root over (N)} points, where √{square root over (N)} is aninteger, but are separated by one or more regions. Thus:

$\begin{matrix}{{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( \frac{1}{N^{2}} \right)\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = {1 - \sqrt{N}}}^{\sqrt{N} - 1}{{k\left( {\sqrt{N} - i} \right)}\left( {\sqrt{N} - {j}} \right)\rho^{d}}}} \right\rbrack}} & (73)\end{matrix}$

where:

-   -   1, when i=0

k=

-   -   2, when i>0, and

${d = {\left( \sqrt{i^{2} + \left( {j + {M\sqrt{N}}} \right)^{2}} \right)\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right)}},{and}$

such that M equals the number of regions.

FIG. 17 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions. FIG. 17suggests that the probability density function can be estimated usingthe following distribution:

$\begin{matrix}{f = \left\{ \begin{matrix}{{1 - \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)}\mspace{11mu}} & {\;{{{{for}\mspace{14mu}{Spacing}} - \sqrt{Area}} \leq D \leq {Spacing}}} \\{{1 + \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)}\mspace{14mu}} & {{{for}\mspace{14mu}{Spacing}} \leq D \leq {{Spacing} + \sqrt{Area}}} \\0 & {{all}\mspace{14mu}{else}}\end{matrix} \right.} & (74)\end{matrix}$

This function is a probability density function because the integrationover all possible values equals zero. FIG. 18 is a graph depicting, byway of example, results by application of this model.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

1. A computer-implemented system for correlating satellite imagery foruse in photovoltaic fleet output estimation, comprising: a computercomprising a processor and memory within which code for execution by theprocessor is stored, comprising: a pixel correlation module configuredto correlate a set of pixels in satellite imagery data of overhead skyclearness to a bounded area within a geographic region, each pixelrepresenting collective irradiance over a plurality of points within thebounded area, each point suitable for operation of a photovoltaic systemcomprised in a photovoltaic fleet; an averaging module configured toconvert the collective irradiance into a set of point clearness indexesfor each of the points within the bounded area relative to clear skyirradiance, and to average the point clearness indexes in the set ofpoint clearness indexes for each of the points within the bounded areainto an area clearness index; an area variance module configured todetermine a variance of the area clearness index in proportion to aphysical area within the bounded area covered by each pixel; and a pointvariance module configured to determine, for each of the points withinthe bounded area, a variance of the point clearness index for that pointas a ratio of the variance of the area clearness index and the physicalarea relative to the point clearness index for that point, cloud speedover the geographic region, and a time interval relating to a timeresolution of observation for the collective irradiance.
 2. A systemaccording to claim 1, further comprising: an area averaging moduleconfigured to evaluate an area function for each pixel by solving adiscrete correlation coefficient matrix comprising correlationcoefficients between point clearness indexes selected for pairs of thepoints in a satellite pixel, and to set the collective irradiance forthe pixel as the solution to the discrete correlation coefficientmatrix.
 3. A system according to claim 1, further comprising: an areaaveraging module configured to evaluate an area function for each pixelby solving probability density function based on a distance for pairs ofthe points in a satellite pixel comprising solving an integral of theprobability density function for the distance as a multiple of acorrelation coefficient function at the distance, and set the collectiveirradiance for the pixel as the solution to the probability densityfunction.
 4. A system according to claim 3, wherein the correlationcoefficient function comprises an exponentially decaying function.
 5. Asystem according to claim 1, further comprising: a fleet statisticsmodule configured to convert the collective irradiance into a set ofpoint sky clearness indexes for each of the points within the boundedarea relative to clear sky irradiance, and to interpret the set of pointsky clearness indexes for each point as area irradiance statistics andassembling the area irradiance statistics for all of the points withinthe bounded area into fleet irradiance statistics applicable over thegeographic region.
 6. A system according to claim 5, further comprising:an average point statistics module configured to convert the areairradiance statistics for each of the points within the bounded areainto irradiance statistics for an average point within the set ofpixels, and to interpret the irradiance statistics for the average pointas average point statistics, each comprising an average of all areairradiance statistics for the set of pixels.
 7. A system according toclaim 1, further comprising: a stored time period relating to a timeresolution of the measuring of the collective irradiance for each pixelin the set of pixels; a time series module configured to divide the setof pixels into a time series by increments of the time period, and toform a time series set of the output sky clearness indexes over eachtime period increment in the time series; and a statistics moduleconfigured to determine irradiance statistics from the time series setof the output sky clearness indexes at each location in the geographicregion.
 8. A system according to claim 1, further comprising: a fleetstatistics module configured to determine fleet irradiance statisticsfor the photovoltaic fleet through statistical evaluation of the skyclearness index and the variance of the sky clearness index for eachpoint within the bounded area, and to build power statistics for thephotovoltaic fleet as a function of the fleet irradiance statistics anda power rating of the photovoltaic fleet.
 9. A system according to claim8, further comprising: a stored time increment relating to a timeresolution of the power statistics for the photovoltaic fleet; a timelag correlation module configured to express a time lag correlationcoefficient as a relationship between the power statistics for thephotovoltaic fleet starting at a beginning of a time period and thepower statistics for the photovoltaic fleet starting at the beginning ofthe time period plus the time increment, and to apply the time lagcorrelation coefficient to the power statistics for the photovoltaicfleet over successive time increments to form a time series of the powerstatistics for the photovoltaic fleet.
 10. A system according to claim9, further comprising: a stored power output for the photovoltaic fleetas part of the power statistics; and a time lag correlation modulefurther configured to express the time lag correlation coefficient interms of the power rating of the photovoltaic fleet and the fleetirradiance statistics.
 11. A computer-implemented method for correlatingsatellite imagery for use in photovoltaic fleet output estimation,comprising the steps of: correlating a set of pixels in satelliteimagery data of overhead sky clearness to a bounded area within ageographic region, each pixel representing collective irradiance over aplurality of points within the bounded area, each point suitable foroperation of a photovoltaic system comprised in a photovoltaic fleet;converting the collective irradiance into a set of point clearnessindexes for each of the points within the bounded area relative to clearsky irradiance; averaging the point clearness indexes in the set ofpoint clearness indexes for each of the points within the bounded areainto an area clearness index; determining a variance of the areaclearness index in proportion to a physical area within the bounded areacovered by each pixel; and for each of the points within the boundedarea, determining a variance of the point clearness index for that pointas a ratio of the variance of the area clearness index and the physicalarea relative to the point clearness index for that point, cloud speedover the geographic region, and a time interval relating to a timeresolution of observation for the collective irradiance, wherein thesteps are performed on a suitably-programmed computer.
 12. A methodaccording to claim 11, further comprising the steps of: evaluating anarea function for each pixel by solving a discrete correlationcoefficient matrix comprising correlation coefficients between pointclearness indexes selected for pairs of the points in a satellite pixel;and setting the collective irradiance for the pixel as the solution tothe discrete correlation coefficient matrix.
 13. A method according toclaim 11, further comprising the steps of: evaluating an area functionfor each pixel by solving probability density function based on adistance for pairs of the points in a satellite pixel comprising solvingan integral of the probability density function for the distance as amultiple of a correlation coefficient function at the distance; andsetting the collective irradiance for the pixel as the solution to theprobability density function.
 14. A method according to claim 13,wherein the correlation coefficient function comprises an exponentiallydecaying function.
 15. A method according to claim 11, furthercomprising the steps of: converting the collective irradiance into a setof point sky clearness indexes for each of the points within the boundedarea relative to clear sky irradiance; and interpreting the set of pointsky clearness indexes for each point as area irradiance statistics andassembling the area irradiance statistics for all of the points withinthe bounded area into fleet irradiance statistics applicable over thegeographic region.
 16. A method according to claim 15, furthercomprising the steps of: converting the area irradiance statistics foreach of the points within the bounded area into irradiance statisticsfor an average point within the set of pixels; and interpreting theirradiance statistics for the average point as average point statistics,each comprising an average of all area irradiance statistics for the setof pixels.
 17. A method according to claim 11, further comprising thesteps of: selecting a time period relating to a time resolution of themeasuring of the collective irradiance for each pixel in the set ofpixels; dividing the set of pixels into a time series by increments ofthe time period; forming a time series set of the output sky clearnessindexes over each time period increment in the time series; anddetermining irradiance statistics from the time series set of the outputsky clearness indexes at each location in the geographic region.
 18. Amethod according to claim 11, further comprising the steps of:determining fleet irradiance statistics for the photovoltaic fleetthrough statistical evaluation of the sky clearness index and thevariance of the sky clearness index for each point within the boundedarea; and building power statistics for the photovoltaic fleet as afunction of the fleet irradiance statistics and a power rating of thephotovoltaic fleet.
 19. A method according to claim 18, furthercomprising the steps of: selecting a time increment relating to a timeresolution of the power statistics for the photovoltaic fleet;expressing a time lag correlation coefficient as a relationship betweenthe power statistics for the photovoltaic fleet starting at a beginningof a time period and the power statistics for the photovoltaic fleetstarting at the beginning of the time period plus the time increment;and applying the time lag correlation coefficient to the powerstatistics for the photovoltaic fleet over successive time increments toform a time series of the power statistics for the photovoltaic fleet.20. A method according to claim 19, further comprising the steps of:determining a power output for the photovoltaic fleet as part of thepower statistics; and expressing the time lag correlation coefficient interms of the power rating of the photovoltaic fleet and the fleetirradiance statistics.
 21. A non-transitory computer readable storagemedium storing code for executing on a computer system to perform themethod according to claim
 11. 22. A computer-implemented apparatus forcorrelating satellite imagery for use in photovoltaic fleet outputestimation, comprising: means for correlating a set of pixels insatellite imagery data of overhead sky clearness to a bounded areawithin a geographic region, each pixel representing collectiveirradiance over a plurality of points within the bounded area, eachpoint suitable for operation of a photovoltaic system comprised in aphotovoltaic fleet; means for converting the collective irradiance intoa set of point clearness indexes for each of the points within thebounded area relative to clear sky irradiance; means for averaging thepoint clearness indexes in the set of point clearness indexes for eachof the points within the bounded area into an area clearness index;means for determining a variance of the area clearness index inproportion to a physical area within the bounded area covered by eachpixel; and means for determining for each of the points within thebounded area a variance of the point clearness index for that point as aratio of the variance of the area clearness index and the physical arearelative to the point clearness index for that point, cloud speed overthe geographic region, and a time interval relating to a time resolutionof observation for the collective irradiance.